more waves

Question 1

speed of sound in gasses equation.

Answer 1

v = [ γP₀/ρ ]^½
P₀: equilibrium pressure
γ: adiabatic index
ρ: gas density

Question 2

2D wave equation

Answer 2

f is a function f(x,y,t) :

∇f² = v⁻² * ∂f²/∂t²

where v= (T/σ)^½
σ: density (per unit area)
T: tension

Question 3

reflection and transmission coefficients range of values

Answer 3

-1 < R < 1

T = R +1
hence: 0 < T < 2

Question 4

Diffraction equations

Answer 4

Single slit: (path difference equation)
D*sin(θ) = nλ
D: slit width
θ: angle to the minima/maxima
n: integer for minima, integer + 0.5 for maxima
The central maxima is at n=0, the first minima is at n=1, hence the first maxima is at n=1.5

Double slit:
d*sin(θ) = nλ
d: slit separation

Question 5

when given a graph of a decaying wave, how do you determine the quality factor

Answer 5

find the frequency from the time period. f= 1/T, ω= 2πf
find the decay constant using I = I₀ * e^(-γt/2)
Q = ω/γ

Question 6

why is SHM in everything

Answer 6

define an arbitrary potential as U(x)

Maclaurin series expand that:
U(x) ≈ U₀ + x(du/dx)ₓ₌₀ + ½x² (d²U/dx²)ₓ₌₀ + …

define x=0 to be the equilibrium point, hence (dU/dx))ₓ₌₀ = 0
since U is an additive property, U₀ = 0
hence: U(x) ≈ ½x² (d²U/dx²)ₓ₌₀ (assuming x is small enough to ignore higher terms)

(d²U/dx²)ₓ₌₀ is a constant, so x² is the only variable term

F = -dU/dx ≈ -x*(d²U/dx²)ₓ₌₀
hence F ∝ -x
∴ a ∝ -x which defines SHM

Question 7

what are the normal frequencies of oscillations?

Answer 7

the square root of the eigenvalues of the 2nd order equations when using the trial solution x = X*e^(iΩt)

Question 8

what do the differential equations look like for a spring with a mass attached to the ceiling look like?

Answer 8

__
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O
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O
(looks like this)

mẍ₁ = -k₁x₁ + k₂(x₂-x₁)
x₁: the displacement of the top ball

the weight is ignored and the upward force is taken as negative (the force driving the oscillations)

Question 9

how would you compare the amplitude and phase of displacement of each mass in a coupled oscillator

Answer 9

amplitude: sub in the eigenvalues calculated to get X₁ in terms of X₂.
this also gives the phase for each of the eigenvalues (normal frequencies). If X₁ = -X₂ they are out of phase

Also, calculate the eigenvectors